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point P traverses CD, while Q simultaneously traverses AE,
and each point of CD has one, and only one, 'paired** point of
An even more unexpected result can be proved. Any line-
segment 5 no matter how small, contains as many points as an
infinite straight line. Further, the segment contains as many
points as there are in an entire plane, or in the whole of three-
dimensional space, or in the whole of space of n dimensions
(where n is any integer greater than zero) or, finally, in a space
of a denumerably infinite number of dimensions.
In all this we have not yet attempted to define a class or a set.
Possibly (as Russell held in 1912) it is not necessary to do so in
order to have a clear conception of Cantor's theory or for that
theory to be consistent with itself - which is enough to demand
of any mathematical theory. Nevertheless present disputes
seem to require that some clear9 self-consistent definition be
given. The following used to be thought satisfactory.
A set is characterized by three qualities: it contains all things
iio which a certain definite property (say redness, or volume, or
taste) belongs; no thing not having this property belongs to the
set; each thing in the set is recognizable as the same thing and
as different from all other things in the set - briefly, each thing
hi the set has a permanently recognizable individuality. The
set itself is to be grasped as a whole. This definition may be too
drastic for use. Consider, for example, what happens to Cantor's
set of all transcendental numbers tinder the third demand*
At this point we may glance back over the whole history of
mathematics - or as much of it as is revealed by the treatises of
the master mathematicians in their purely technical works -
and note two modes of expression which recur constantly in
nearly all mathematical exposition. The reader perhaps hag
been irritated by the repetitious use of phrases such as fiwe can
find a whole number greater than 2% or *we can choose a number
less than n and greater than n  2.' The choice of such phrase-
ology is not merely stereotyped pedantry. There is a reason for
its use, and careful writers mean exactly what they say when
they assert that Sre can find, etc*. They mean that they can do
what they say.